Learn Hardcore Spanish (Spain): En la universidad estudio filosofía, pero mi mejor amiga prefiere biología. - At university I study philosophy, but my best friend prefers biology.

You also need the article la because Spanish almost always uses an article with universidad in this kind of sentence. English can say at university with no article, but in Spanish the natural phrase is en la universidad, not en universidad.

Can I change the word order and say Estudio filosofía en la universidad?

Yes. Both are correct:

En la universidad estudio filosofía.
Estudio filosofía en la universidad.

Spanish word order is flexible. Putting en la universidad at the beginning slightly emphasizes the place (at university, this is what I study). Putting it at the end feels a bit more neutral, like English. Grammatically, both are fine.

Why is there no yo before estudio and no ella before prefiere?

Spanish often drops subject pronouns because the verb ending already shows who the subject is:

estudio = I study (first person singular)
prefiere = he/she prefers (third person singular)

You add yo, él, ella, etc. only when you want to emphasize or clarify:

Yo estudio filosofía, pero ella prefiere biología. (I do X, but she prefers Y.)

In the original sentence, the context makes the subjects obvious, so the pronouns are omitted.

Why is it estudio filosofía and not estoy estudiando filosofía?

Spanish uses the simple present much more than English to talk about ongoing or regular activities:

Estudio filosofía = I study / I am studying philosophy (in general, as my degree/program).
Estoy estudiando filosofía is more like I am in the middle of studying philosophy right now, or temporarily, and is less common to describe your major.

To say what you study at university as a general fact, estudio filosofía is the normal choice.

Why is there no article before filosofía and biología?

With school subjects and academic fields, Spanish often omits the article after certain verbs, especially estudiar and similar:

Estudio filosofía.
Mi amiga prefiere biología.
Quiero hacer medicina.

You would usually not say estudio la filosofía here.

However, you can use the article when you talk about the subject more abstractly as a discipline, not as your course of study:

La filosofía es una disciplina muy antigua. (Philosophy as a field in general)
La biología estudia los seres vivos.

So is prefiere la biología wrong?

It is not wrong; it just sounds a bit different.

Prefiere biología usually sounds like “she prefers biology (as a subject to study).”
Prefiere la biología can sound more like “she prefers biology (as a discipline/field),” or you are contrasting it with another specific discipline:
Prefiere la biología a la filosofía.

In everyday conversation about school subjects in Spain, prefiere biología is very natural and common.

Why is it mi mejor amiga and not mi amiga mejor?

Most adjectives come after the noun in Spanish (e.g. casa grande, libro interesante), but some very common ones almost always go before the noun, including:

mejor (better / best)
peor (worse / worst)
mayor, menor in many uses

That’s why:

mi mejor amiga = my best friend
not mi amiga mejor.

Mi amiga mejor is not idiomatic for “my best friend”. The usual expression is mi mejor amigo / amiga.

Why is it mejor amiga and not something like amiga más buena for “best friend”?

For best in the sense of best X, Spanish normally uses mejor:

mi mejor amiga = my best (female) friend
mi mejor amigo = my best (male) friend

más bueno/a usually means nicer / kinder / more good (moral quality or behavior), not best in this structural sense. So:

Ella es más buena = She is kinder / a better person.
Ella es mi mejor amiga = She is my best friend.

Why is amiga feminine? What if my best friend is a man?

Spanish nouns that refer to people often change endings for gender:

amigo = male friend
amiga = female friend

So:

mi mejor amiga = my best friend (female)
mi mejor amigo = my best friend (male)

If you are talking about a mixed or unspecified group, you might see amigos as the default plural in traditional grammar:
mis mejores amigos (my best friends).

Why is pero used and not sino?

Both pero and sino can translate as but, but they are used differently:

pero = but / however. It links two ideas, where the second contrasts with or limits the first.
Estudio filosofía, pero mi mejor amiga prefiere biología.
sino = but rather / but instead. It is used after a negation, to correct or replace what came before:
No estudio filosofía, sino biología.
(I don’t study philosophy, but rather biology.)

There is no negation in the original sentence, so you must use pero, not sino.

Why is it prefiere and not prefiero?

The verb preferir (to prefer) is conjugated in the present tense as:

yo prefiero – I prefer
tú prefieres – you prefer
él / ella / usted prefiere – he / she / you (formal) prefer(s)

The subject here is mi mejor amiga (my best friend), so you need the third person singular form:

Mi mejor amiga prefiere biología. = My best friend prefers biology.

Why do filosofía and biología have an accent on -ía?

Spanish accent marks show where the stress falls when it does not follow the normal rules.

Words ending in a vowel, n, or s are normally stressed on the second-to-last syllable. Without an accent:

filosofia would be stressed fi-lo-SO-fia
biologia would be bio-LO-gia

But we actually pronounce them:

fi-lo-so-FÍ-a
bio-lo-GÍ-a

The written accents on í in filosofía and biología mark this irregular stress so that the written form matches the spoken form.